{
  "abstract": "Let $M$ be a symmetric matrix over $\\mathbb{F}_2$, and let $\\operatorname{diag}(M)$ be its diagonal vector. It is known that $\\operatorname{diag}(M)\\in\\operatorname{Im}(M)$. Thus the affine system $Mx=\\operatorname{diag}(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies $\\operatorname{diag}(M)^T x\\equiv\\operatorname{rank}(M)\\pmod{2}$. For graph matrices, this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$. We also study how rank and nullity change when loops are toggled. Algebraically, simultaneous loop toggling on the support of a vector $u$ is the diagonal rank-one update $M\\mapsto M+uu^T$. We prove an exact three-case rank and nullity formula for this update. Finally, for rooted trees with arbitrary binary diagonal labels, we give a finite-state boundary recursion using affine subspaces of $\\mathbb F_2^2$. This recursion counts all generalized odd-domination patterns and implies eventual quasigeometric nullity formulas for complete rooted trees with eventually periodic depth labels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.11056",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohsen Aliabadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vzye3fcsm8p9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.11056",
  "fulltext_url": null,
  "id": "9vzye3fcsm8p9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.11056",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7161014364",
  "pdf_url": "https://arxiv.org/pdf/2605.11056",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vzye3fcsm8p9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.11056",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.11056"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Diagonal parity and loop toggling for symmetric matrices over $\\mathbb F_2$",
  "title_text": "Diagonal parity and loop toggling for symmetric matrices over 𝔽_2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/9vzye3fcsm8p9",
  "venue": "arXiv",
  "versions": []
}